Hilbert schemes, polygraphs and the Macdonald positivity conjecture

Abstract

We study the isospectral Hilbert scheme X n X_{n} , defined as the reduced fiber product of ( C 2 ) n (\mathbb {C}^{2})^{n} with the Hilbert scheme H n H_{n} of points in the plane C 2 \mathbb {C}^{2} , over the symmetric power S n C 2 = ( C 2 ) n / S n S^{n}\mathbb {C}^{2} = (\mathbb {C}^{2})^{n}/S_{n} . By a theorem of Fogarty, H n H_{n} is smooth. We prove that X n X_{n} is normal, Cohen-Macaulay and Gorenstein, and hence flat over H n H_{n} . We derive two important consequences. (1) We prove the strong form of the n ! n! conjecture of Garsia and the author, giving a representation-theoretic interpretation of the Kostka-Macdonald coefficients K λ μ ( q , t ) K_{\lambda \mu }(q,t) . This establishes the Macdonald positivity conjecture , namely that K λ μ ( q , t ) ∈ N [ q , t ] K_{\lambda \mu }(q,t)\in {\mathbb N} [q,t] . (2) We show that the Hilbert scheme H n H_{n} is isomorphic to the G G -Hilbert scheme ( C 2 ) n / / S n (\mathbb {C}^{2})^{n}{/\!\!/}S_n of Nakamura, in such a way that X n X_{n} is identified with the universal family over ( C 2 ) n / / S n ({\mathbb C}^2)^n{/\!\!/}S_n . From this point of view, K λ μ ( q , t ) K_{\lambda \mu }(q,t) describes the fiber of a character sheaf C λ C_{\lambda } at a torus-fixed point of ( C 2 ) n / / S n ({\mathbb C}^2)^n{/\!\!/}S_n corresponding to μ \mu . The proofs rely on a study of certain subspace arrangements Z ( n , l ) ⊆ ( C 2 ) n + l Z(n,l)\subseteq (\mathbb {C}^{2})^{n+l} , called polygraphs , whose coordinate rings R ( n , l ) R(n,l) carry geometric information about X n X_{n} . The key result is that R ( n , l ) R(n,l) is a free module over the polynomial ring in one set of coordinates on ( C 2 ) n (\mathbb {C}^{2})^{n} . This is proven by an intricate inductive argument based on elementary commutative algebra.